Using your Head is Permitted

December 2014 riddle

This month's riddle is a follow-up from last month
and the month before it. It was suggested by
Radu-Alexandru Todor and Lorenzo Gianferrari Pini. (Thanks!)

Spoiler alert: The following description contains hints
regarding the solutions of the October and November riddles. If you still mean
to tackle those other riddles, you may want to hold on before you continue to
this one.

This month's riddle comes in two parts. Answer either or both parts.
Separate solver lists will be kept for each part.

Part 1:

Let a "casino" be a point, p, on the interval [0,1], such that for any
integer n≥2 and for any open neighbourhood N of p
there is a point pn in N such that the
n-sided dice can be simulated by a biased coin with probability
pn in a finite number of throws.

Characterise exactly (with proof) the subset of
[0,1] that is the set of casinos.

Part 2:

Let a "rational casino" be a point (not necessarily rational-valued),
(p,q), in the set [0,1]2, such that for any integer
n≥2 and any open neighbourhood N of (p,q),
there is a rational-valued point
(pn,qn) in N, such that a pair of
biased coins with probabilities pn and qn,
respectively, can simulate an n-sided dice in a finite number of throws.

Find (with proof) one such rational casino.

An asterisk will be given to solvers able to characterise exactly the subset of
[0,1]2 that is the set of rational casinos.